p-adic空间上分布可积性定理的部分类比及其应用
A partial analog of integrability theorem for distributions on p-adic spaces and applications
摘要：
设X是光滑实代数簇.设$ξ$是其上的一个分布，可以定义$ξ$的奇异支集为由$ξ$生成的$D_X$ -模的奇异支集（有时也称为特征簇）。奇异支集的一个强有力的性质是它是$T^*X$的余各向同性子簇。这是可积性定理（见[KKS，马尔，加布]）。这个定理原来是有用的表示论的真正约化群（见例如[AG_AMOT，AS，说]）。本文的目的是给这个定理的一个模拟的非阿基米德的情况下。由于D-模的理论对我们来说是不可用的，所以我们需要一个不同的奇异支撑的定义。我们使用的概念波前集从[Hef]和定义的奇异支持是它的Zapriki封闭。然后我们证明了奇异支撑满足一些性质，我们称之为弱共各向同性，这是弱于共各向同性，但对于某些应用是足够的。我们还证明了其他一些性质的奇异支持是平凡的阿基米德的情况下（使用代数定义），但不明显的非阿基米德的情况下。
英文摘要：
Let X be a smooth real algebraic variety. Let $ξ$ be a distribution on it. One can define the singular support of $ξ$ to be the singular support of the $D_X$-module generated by $ξ$ (some times it is also called the characteristic variety). A powerful property of the singular support is that it is a coisotropic subvariety of $T^*X$. This is the integrability theorem (see [KKS, Mal, Gab]). This theorem turned out to be useful in representation theory of real reductive groups (see e.g. [AG_AMOT, AS, Say]). The aim of this paper is to give an analog of this theorem to the non-Archimedean case. The theory of D-modules is not available to us so we need a different definition of the singular support. We use the notion wave front set from [Hef] and define the singular support to be its Zariski closure. Then we prove that the singular support satisfies some property that we call weakly coisotropic, which is weaker than being coisotropic but is enough for some applications. We also prove some other properties of the singular support that were trivial in the Archimedean case (using the algebraic definition) but not obvious in the non-Archimedean case.
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